Lines of Symmetry & Rotational Symmetry

Lines of Symmetry

Definition: A line of symmetry is a line that divides a shape into two parts that are mirror images of each other. If you fold the shape along this line, both halves match exactly.

Lines of symmetry are always straight lines. They can be vertical, horizontal, or diagonal depending on the shape.

Identifying Lines of Symmetry in 2D Shapes

To find lines of symmetry, imagine folding the shape along a line so that one side fits perfectly over the other. The line where this happens is a line of symmetry.

For instance, a square has 4 lines of symmetry: 2 lines through the midpoints of opposite sides (vertical and horizontal) and 2 lines through opposite corners (diagonals).

Other shapes have different numbers of lines of symmetry:

• Equilateral triangle: 3 lines of symmetry, each passing through a vertex and the midpoint of the opposite side.
• Rectangle (not square): 2 lines of symmetry, vertical and horizontal through midpoints of opposite sides.
• Isosceles triangle: 1 line of symmetry, vertical through the vertex angle and midpoint of the base.
• Circle: Infinite lines of symmetry, any line through the centre.
• Regular polygons: Number of lines of symmetry equals the number of sides.

Lines of Symmetry in Regular Polygons

A regular polygon is a polygon with all sides and angles equal. These shapes have symmetrical properties:

• The number of lines of symmetry is equal to the number of sides.
• Each line of symmetry passes through a vertex and the midpoint of the opposite side (for polygons with an even number of sides, some lines pass through opposite vertices).

For example:

• Regular pentagon: 5 lines of symmetry
• Regular hexagon: 6 lines of symmetry
• Regular octagon: 8 lines of symmetry

Irregular polygons usually have fewer or no lines of symmetry.

For instance, a regular hexagon has 6 lines of symmetry: 3 lines pass through opposite vertices, and 3 lines pass through midpoints of opposite sides.

Example: A regular hexagon has 6 lines of symmetry because it has 6 equal sides and angles, and each line divides it into two identical halves.

Rotational Symmetry

Definition: A shape has rotational symmetry if it looks exactly the same after being rotated (turned) about its centre by an angle less than $360^\circ$.

The shape can be rotated clockwise or anticlockwise.

Order of Rotational Symmetry

The order of rotational symmetry is the number of times a shape fits onto itself during one full $360^\circ$ rotation.

For example:

• If a shape looks the same 4 times during a full turn, its order of rotational symmetry is 4.
• If it only looks the same once (at $360^\circ$), the order is 1 (no rotational symmetry).

Finding the Angle of Rotation

The angle of rotation for one step of symmetry is found by dividing $360^\circ$ by the order of rotational symmetry:

$$\text{Angle of rotation} = \frac{360^\circ}{\text{order of rotational symmetry}}$$

This angle tells you how far to rotate the shape so that it looks the same again.

Examples with Common Shapes

• Square: Order 4 rotational symmetry. It fits onto itself 4 times during a full turn at angles $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.
• Equilateral triangle: Order 3 rotational symmetry. It fits onto itself 3 times at $120^\circ$, $240^\circ$, and $360^\circ$.
• Regular pentagon: Order 5 rotational symmetry. Fits onto itself 5 times at $72^\circ$ intervals.
• Rectangle (not square): Order 2 rotational symmetry. Fits onto itself twice at $180^\circ$ and $360^\circ$.
• Circle: Infinite order of rotational symmetry; looks the same at any angle of rotation.

Example: A regular hexagon has order 6 rotational symmetry because it fits onto itself 6 times during a full turn, every $60^\circ$.